Testing Hereditary Properties of Ordered Graphs and Matrices

For classes ${\cal O}$ of structures on finite linear orders (permutations, ordered graphs etc.) endowed with containment order $\preceq$ (containment of permutations, subgraph relation etc.), we investigate restrictions on the function $f(n)$ counting objects with size $n$ in a lower ideal in $({\cal O},\preceq)$. We present a framework of edge $P$-colored complete graphs $({\cal C}(P),\preceq)$ which includes many of these situations, and we prove for it two such restrictions (jumps in growth): $f(n)$ is eventually constant or $f(n)\ge n$ for all $n\ge 1$; $f(n)\le n^c$ for all $n\ge 1$ for a constant $c>0$ or $f(n)\ge F_n$ for all $n\ge 1$, $F_n$ being the Fibonacci numbers. This generalizes a fragment of a more detailed theorem of Balogh, Bollobás and Morris on hereditary properties of ordered graphs.

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Hereditary properties of partitions, ordered graphs and ordered hypergraphs

European Journal of Combinatorics ◽

10.1016/j.ejc.2006.05.004 ◽

2006 ◽

Vol 27 (8) ◽

pp. 1263-1281 ◽

Cited By ~ 8

Author(s):

József Balogh ◽

Béla Bollobás ◽

Robert Morris

Keyword(s):

Ordered Graphs ◽

Hereditary Properties

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Hereditary Properties of Ordered Graphs

Algorithms and Combinatorics - Topics in Discrete Mathematics ◽

10.1007/3-540-33700-8_12 ◽

2007 ◽

pp. 179-213 ◽

Cited By ~ 8

Author(s):

József Balogh ◽

Béla Bollobás ◽

Robert Morris

Keyword(s):

Ordered Graphs ◽

Hereditary Properties

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On spaces in which countably compact sets are closed, and hereditary properties

Topology and its Applications ◽

10.1016/0166-8641(80)90027-9 ◽

1980 ◽

Vol 11 (3) ◽

pp. 281-292 ◽

Cited By ~ 18

Author(s):

Mohammad Ismail ◽

Peter Nyikos

Keyword(s):

Compact Sets ◽

Countably Compact ◽

Hereditary Properties

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Minimal Characteristic Algebras for Rectangular k-Normal Identities

Algebra Colloquium ◽

10.1142/s1005386711000460 ◽

2011 ◽

Vol 18 (04) ◽

pp. 611-628

Author(s):

K. Hambrook ◽

S. L. Wismath

Keyword(s):

Hereditary Property ◽

Fixed Type ◽

Hereditary Properties

A characteristic algebra for a hereditary property of identities of a fixed type τ is an algebra [Formula: see text] such that for any variety V of type τ, we have [Formula: see text] if and only if every identity satisfied by V has the property p. This is equivalent to [Formula: see text] being a generator for the variety determined by all identities of type τ which have property p. Płonka has produced minimal (smallest cardinality) characteristic algebras for a number of hereditary properties, including regularity, normality, uniformity, biregularity, right- and leftmost, outermost, and external-compatibility. In this paper, we use a construction of Płonka to study minimal characteristic algebras for the property of rectangular k-normality. In particular, we construct minimal characteristic algebras of type (2) for k-normality and rectangularity for 1 ≤ k ≤ 3.

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